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Given the integral equation

$$\exp(x)-1=\int_0^{\infty} \frac{\mathrm dt}{t}\operatorname{frac}\left(\frac{ \sqrt x}{\sqrt t}\right) f(t)\;,$$

where $\operatorname{frac}$ denotes the fractional part of a number, $ \operatorname{frac}(x)= x-\lfloor x\rfloor$.

My questions are:

  1. Can we deduce from this integral equation that $ f(x)= O(x^{1/4+\epsilon}) $ for some positive $\epsilon$?

  2. Can we solve this integral by the Hilbert-Schmidt method?

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Some $\TeX$ hints: Use \exp instead of exp to keep the function name from being interpreted as individual variables whose symbols get italicized. If there is no predefined command sequence, e.g. for $\operatorname{frac}$, use \operatorname{frac}. Displayed equations should be in double dollar signs (as opposed to single dollar signs) to allow the proper font sizes for displayed equations to be selected. You can right-click on any $\TeX$ output you see on this site and select "Show Source" to see how it's done. – joriki Oct 15 '11 at 9:40
You might consider the substitution $u=\log t$, then it turns into a convolution on the $\mathbb{R}$. However, I can not see how that would help - also I think Phira might be right. – AD. Oct 15 '11 at 13:50
Why it is obvious that a solution does exist ? – Sasha Oct 15 '11 at 16:06
up vote 1 down vote accepted

You cannot conclude anything for the asymptotics of the integrand, because the function can have very high, very narrow peaks that contribute almost nothing to the integral.

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